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Omni-MATH / A blackboard contains 68 pairs of nonzero integers. Suppose that for each…

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problem

A blackboard contains 68 pairs of nonzero integers. Suppose that for each positive integer kk at most one of the pairs (k,k)(k, k) and (−k,−k)(-k, -k) is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student then scores one point for each of the 68 pairs in which at least one integer is erased. Determine, with proof, the largest number NN of points that the student can guarantee to score regardless of which 68 pairs have been written on the board.
Plain-text mathematical notation (without MathML)
A blackboard contains 68 pairs of nonzero integers.  Suppose that for each positive integer k at most one of the pairs (k,k) and (−k,−k) is written on the blackboard.  A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0.  The student then scores one point for each of the 68 pairs in which at least one integer is erased.  Determine, with proof, the largest number N of points that the student can guarantee to score regardless of which 68 pairs have been written on the board.
Original LaTeX notation
A blackboard contains 68 pairs of nonzero integers.  Suppose that for each positive integer $k$ at most one of the pairs $(k, k)$ and $(-k, -k)$ is written on the blackboard.  A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0.  The student then scores one point for each of the 68 pairs in which at least one integer is erased.  Determine, with proof, the largest number $N$ of points that the student can guarantee to score regardless of which 68 pairs have been written on the board.

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Source and history

task_id
9ac6dbf1-d423-5d46-92db-f383c0cd501a
task_key
test--9ac6dbf1-d423-5d46-92db-f383c0cd501a
task_revision_id
1
config / split
/ test

Official source

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